Method and system for long range wireless power transfer

ABSTRACT

A wireless energy transfer system includes a primary and one (or more) secondary oscillators for transferring energy therebetween when resonating at the same frequency. The long range (up to and beyond 100 m) efficient (as high as and above 50%) energy transfer is achieved due to minimizing (or eliminating) losses in the system. Superconducting materials are used for all current carrying elements, dielectrics are either avoided altogether, or those are used with a low dissipation factor, and the system is operated at reduced frequencies (below 1 MHz). The oscillators are contoured as a compact flat coil formed from a superconducting wire material. The energy wavelengths exceed the coils diameter by several orders of magnitude. The reduction in radiative losses is enhanced by adding external dielectric-less electrical capacitance to each oscillator coil to reduce the operating frequency. The dielectric strength of the capacitor is increased by applying a magnetic cross-field to the capacitor to impede the electrons motion across an air gap defined between coaxial cylindrical electrodes.

REFERENCE TO THE RELATED APPLICATION

This utility patent application is based on the Provisional PatentApplication No. 61/350,229 filed 1 Jun. 2010.

FIELD OF THE INVENTION

The present invention is directed to power transfer systems, and more inparticular to a system for wireless energy transfer betweenelectromagnetic resonant objects.

More in particular, the present invention is directed to the powertransfer between resonators (also referred to herein intermittently asoscillators) in highly efficient and low loss manner, thereby permittinglong range wireless power transfer.

The present invention is further directed to a system for wireless powertransfer with diminished resistive, dielectric, and radiative losses,where such is operated at reduced frequencies to produce higherefficiencies at long range distances.

The present invention also is directed to a system for wireless powertransfer between electromagnetic oscillators spaced apart at a desireddistance from each other where the system components are manufacturedfrom superconductive materials for diminishing resistive losses. Use ofdielectric materials is minimized or avoided to decrease the dielectriclosses, and the operating resonant frequency is maintained below apredetermined level to attain a desired amount of power transfer at anincreased efficiency level.

In addition, the present invention is directed to a system for wirelesspower transfer, where the oscillators are formed as superconductivedielectric-less compact (flat) coils coupled to dielectric-less (andpreferably superconductive) capacitors contoured in a shape permittingthe application of a magnetic field for increasing the effectivedielectric strength of air or other medium between the capacitorelectrodes thereby permitting a dielectric-less capacitive element withsatisfactory dielectric properties.

BACKGROUND OF THE INVENTION

Wireless energy (or power) transfer is a promising approach forenvironmentally friendly, convenient and reliable powering of electricaland electronic devices, such as computers, electric vehicles, cellphones, etc.

Resonant Inductive Coupling pioneered by Nikola Tesla in the early20^(th) century has later found applications in power transfer systems.

Recent developments in the field of power transfer have demonstrated theability to transfer 60 W power with 40% efficiency covering 2 mdistance. This medium-range wireless energy transfer system (called“WiTricity”) has been developed by a group of MIT scientists based onstrong coupling between electromagnetic resonant objects, i.e.,transmitters and receivers that contain magnetic loop antennascritically attuned to the same frequency. As presented in A. Karalis, etal., “Efficient Wireless Non-Radiative Mid-Range Energy Transfer”, Ann.Phys., 10.1016 (2007), and U.S. Pat. Nos. 7,741,734 and 7,825,543, thesystem for wireless energy transfer includes a first resonator structureconfigured to transfer energy non-radiatively to a second resonatorstructure over medium range distances. These distances are characterizedas being large in comparison to transmit-receive antennas, but small incomparison to the wavelength of the transmitted power.

The resonators in these energy transfer systems are formed asself-resonant conducting coils from a conductive wire which is woundinto a helical coil of a predetermined radius r and height h surroundedby air, as shown in FIG. 1. The non-radiative energy transfer in thissystem is mediated by a coupling of a resonant field evanescent tail ofthe first resonator structure and a resonant field evanescent tail ofthe second resonator structure.

The ability to effectively transfer power over desired distances,depends on losses in the resonance system which may be attributed toohmic (material absorption) loss inside the wire, radiative loss in thefree space, as well as dielectric losses in dielectric materials used inthe system.

In “WiTricity,” the maximum power coupling efficiency of coilsfabricated from standard conductors occurs at the 10 MHz range, wherethe combination of resistive and radiative losses are at a minimum. Theeffective range of these systems, i.e., a few meters at non-negligibleefficiencies, has significant application within everyday life toprovide power to personal electronics (laptops, cell phones) or otherequipment within the confines of room. However, this type of system isincapable of efficient power transfer with respect to relatively longrange applications.

It would be highly desirable to extend the reach of the resonantinductive power transfer for applications in space, for example, for theon-orbit power transfer between the elements of a satellite cluster oron the surface of the moon between a centralized power station and arover. In order to attain greater distances in wireless power transfer,higher efficiencies of power transfer are necessary. Therefore, it ishighly desirable to provide a long range power transfer system where theloss paths existing in the mid-range system are minimized or eliminated.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to provide a longrange wireless power transfer system where high efficiencies of powertransfer are attained due to a reduction in or elimination of parasiticloss mechanisms attributed to the internal material dissipation(resistive, or ohmic, losses) as well as radiative losses.

It is a further object of the present invention to provide a long rangeinductive power transfer system in which the oscillators aremanufactured, based on superconducting principles (superconductingmaterials, as well as compactness for cryo-cooling) to reduce the ohmiclosses.

It is another object of the present invention to provide an efficientlong range wireless power transfer system in which the system componentsare free of dielectric losses.

It is also an object of the present invention to provide a long rangeinductive wireless power transfer system in which external capacitancesare coupled to the superconductive oscillators to lower the operatingfrequency, thereby attaining higher efficiencies and thus permittingpower transfer over greater distances. Preferably, the capacitors aremanufactured from superconductive materials and are dielectric-less.

It is still a further object of the present invention to provide a powertransfer system using superconducting dielectric-less system componentswhile operating the system at a reduced operating frequency to attaineffective power transfer over extended distances.

In one aspect, the present invention is envisioned as a system for longrange wireless power transfer which comprises a primary oscillator andone (or a plurality of) secondary oscillator(s) displaced from theprimary oscillator at a distance D (which may fall in any desired powertransfer range, including both mid-range, as well as long-range over 100m, for instance) to receive energy from the primary oscillator. Theoscillators are configured into flat compact coils formed from asuperconducting material and resonating substantially at the samefrequency. The frequency is maintained below a predetermined frequencylevel (below 1 MHz, and preferably, at or below ˜200 KHz) which providesa significant reduction in radiative losses in both the primary andsecondary oscillators. It is important that the system is operated atwavelengths that exceed the diameter of the coils by several orders ofmagnitude.

The system includes a source of energy coupled up-stream of said primaryoscillator and one or a plurality of power consuming unit(s) eachcoupled down-stream of the respective secondary oscillator.

A uni-turn drive coil is coupled between the source of energy and theprimary coil. A drain coil is coupled between the secondary oscillatorand the respective power consuming unit.

A plurality of capacitors may be employed in the instant system. Eachcapacitor is coupled to a respective oscillator. The capacitor includesa pair (or more) of coaxially disposed cylindrical electrodes includingan inner cylindrical electrode and one (or more) outer cylindricalelectrode(s) disposed in a co-axial surrounding relationship with theinner cylindrical electrode. An air gap is defined between cylindricalwalls of the inner and outer cylindrical electrodes.

Preferably, the capacitor, similar to the oscillators, is formed from asuperconducting material. The superconducting material for theoscillators and the capacitors may be any superconductor, including forexample, Type I superconductors, High Temperature Superconductors, suchas BSCCO, or YBCO, as well as room temperature superconductors.

Although, the air gap in the capacitor, and spaces between windings incoils may be filled with a dielectric material having a low dissipationfactor, it is preferred that dielectric-less components are used in thesystem. In order to provide dielectric-less capacitor, having asatisfactory dielectric strength, a magnetic field is applied axially tothe capacitor to increase a breakdown voltage threshold in its air gap,thereby increasing the effective dielectric strength of air in the airgap of the dielectric-less capacitor.

A booster resonator coil may be positioned between the primary and thesecondary oscillators to permit even larger transfer distances. Thebooster resonator coil resonates in phase with the primary oscillatorstructure to receive energy from it and is in phase with the secondaryoscillator to transfer power thereto.

A thermo-control system is provided in the subject system, whichcontrols the cryo-equipment operatively coupled to the oscillators andcapacitors. The shape and dimensions of the coils and capacitors must becompatible with dimensional abilities of the cryo-equipment.

The present invention also is envisioned as a method for long rangewireless energy transfer, which includes the steps of:

-   -   fabricating primary and secondary oscillator structures as        compact flat coils formed from a superconducting material,    -   displacing the secondary oscillator from the primary oscillator        a desired distance which may fall in the range below as well as        exceeding 100 m,    -   generating an oscillating current of a resonant frequency in the        primary oscillator so that the oscillating current creates an        oscillating field, and    -   sensing the oscillating current of the primary coil by the        second oscillator, thereby causing oscillation of the secondary        oscillator structure at the same resonant frequency, thus        transferring energy from the primary to the secondary        oscillator.

In the subject method, by maintaining the resonant frequency below apredetermined frequency level (for example, below 1 MHz, and preferablyat or below ˜200 KHz), a reduced radiative loss in both the primary andsecondary oscillators may be attained.

The method further comprises the steps of:

-   -   coupling a capacitor element to each of the primary and        secondary coils where the capacitor is preformed to include an        inner cylindrical electrode and one (or more) outer cylindrical        electrode(s) co-axially disposed around the inner cylindrical        electrode. Specific care is taken to define an air gap between        cylindrical walls of the inner and outer cylindrical electrodes.

Finally, a magnetic field is applied axially to the capacitor toincrease the effective dielectric strength of air in the air gap,thereby permitting formation of a dielectric-less capacitor which,however, has a satisfactory dielectric strength. The capacitorpreferably is formed from a superconducting material.

These and other objects and advantages will become apparent from thefollowing detailed description taken in conjunction with theaccompanying patent Drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1. is a schematic representation of a wireless energy transferscheme of the prior art;

FIG. 2 is a schematic representation of a wireless energy transfersystem of the present invention;

FIG. 3 is an equivalent scheme of the system presented in FIG. 2;

FIG. 4 is a diagram representing the power flow in the present system;

FIG. 5 is a representation of the superconducting compact oscillator ofthe present invention;

FIG. 6 is a representation of a section of the superconductiveoscillator of the present invention with the “holding” mechanism;

FIG. 7 is a perspective view of one configuration of the cylindricalcapacitor used in the present system;

FIG. 8 is a schematic representation of the connection of the capacitorto the superconducting coil;

FIG. 9 shows an alternative implementation of the capacitor of thepresent invention;

FIG. 10 is a diagram representing the variation of the Figure Of Merit(FOM) with dimensionless frequency for different loss mode ratios in thewireless energy transfer system;

FIG. 11 is a diagram representing a maximum efficiency versus normalizedfrequency-distance product for the case of zero ohmic dissipation;

FIG. 12 is a diagram representing efficiency versus power mismatch;

FIG. 13 is a diagram showing the variation in charge state of the inner(Q_(inner)) and outer (Q_(outer)) conductors over a full cycleindicating the notional regions (regions 1, 2) where significantelectron release would typically occur as a result of the electricalfield at the conductor surface;

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring to FIGS. 2 and 3, the system 10 for a long range inductivepower transfer includes a source oscillator 12 (also referred to hereinas a transmitting oscillator, or a primary oscillator) which isconnected to a power source 14, and one or several receiving (alsoreferred to herein as secondary) oscillator(s) 16 physically separatedby a distance D from the source oscillator 12.

As presented in FIG. 2, the system 10 may include a number of receivingoscillators 16, each for powering a corresponding power consuming device18, such as, for example, cell phones, TVs, computers, etc. It is to beunderstood by those skilled in the art, that although any number of thesecondary oscillators 16 is envisioned in the present system, for thesake of simplicity further disclosure will be presented based on asingle secondary oscillator design.

The energy provided by the power source 14 to the source oscillator 12is wirelessly transferred in the system 10 in a non-radiative manner tothe receiving oscillator(s) 16 over the distance D using theelectromagnetic field. The distance D may fall in the range betweenmeters to hundreds (or over) of meters, depending on the application andthe ability of the power transfer system. As will be presented infurther paragraphs, the present system 10 is designed with the abilityto effectively transfer power over long distances, i.e., 100s of meters.

As shown in FIGS. 2-3, and 5-6, the source oscillator 12, i.e., theprimary oscillator, is formed as a multi-winding coil. The receivingoscillator, i.e., the secondary oscillator 16 is similarly formed as amulti-winding coil.

Mounted close to each of the resonant coils, i.e., the primary coil 12and the secondary coil 16 is a single turn coil. The single turn coil 20mounted between the power source 14 and the primary coil 12 is the drivecoil 20. The energy flows from the drive coil 20 to the primary coil 12as shown by the arrow 24. The drive coil 20 is connected to the powersupply 14 for being driven at the natural frequency of the primary coil12. At this frequency, a large oscillating current is generated in theprimary coil 12. This oscillating current creates an oscillatingmagnetic field that is then sensed by the secondary coil 16 which, as aresult, will start oscillating. As the current increases in thesecondary coil 16, more energy will be available for powering therespective power consuming device 18.

A single turn coil 22, referred to herein as the drain coil, couples tothe secondary coil 16. The energy flows from the secondary coil 16 tothe drain coil 22 as shown by the arrow 26. The load, i.e., the powerconsuming device 18, is connected to the drain coil 22 to receive power.

The ends of each resonant coil, i.e., primary coil 12, and secondarycoil 16, may or may not be connected to capacitors. As shown in FIG. 2,the ends 28, 30 of the primary coil 12, as well as the ends 32, 34 ofthe secondary coil 16 are not connected.

In an alternative embodiment, in the equivalent circuit diagram shown inFIG. 3, as well as in FIGS. 8-9, the primary and secondary coilcomponents are each shown connected to a capacitor, i.e., the capacitor36 is coupled to the primary coil 12, and the capacitor 38 coupled tothe secondary coil 16.

Even if the resonant coils 12 and 16 are not connected to the capacitors36 and 38, respectively, they have a “self-capacitance”, which inconjunction with their self inductance causes them to resonate at aparticular frequency. By adding an additional external capacitance, thisfrequency may be lowered, so that it becomes easier to match thefrequencies of the primary and secondary coils 12, 16, respectively. Theoperation of this system 10 is based on the ability to cause resonationof the primary and the secondary coils at the same frequency. By addingadditional capacitances 36, 38 to the primary and secondary coils 12,16, respectively, in addition to matching of the frequency of the coils12 and 16, coils of different sizes may be used. The resonant coils,when fabricated from a superconductive material, as will be presentedfurther herein, may be as small as 10 cm in diameter or as large asseveral meters in diameter. The superconductor material may be selectedfor example, from Type I superconductors, High TemperatureSuperconductors, such as BSCCO, and YBCO, as well as room temperaturesuperconductors.

As long as the two coils, i.e., the primary coil 12 and the secondarycoil 16, have the same resonant frequency, they may be, but do not haveto be of the same physical size. The larger product of the two coilareas, e.g., of the primary coil 12 and the secondary coil 16, leads tothe larger amount of power which can be transferred from the primarycoil 12 to the secondary coil 16 as shown by the arrow 40. In some casesit may be advantageous to increase the diameter of one coil and decreasethe diameter of another coil.

By lowering the losses in the primary coil 12, a higher level of theoscillating current may be created, thereby resulting in a largermagnetic field. Likewise, by lowering the losses in the secondary coil16, a higher level of the oscillating current may be created, and moreenergy will be available to the load device 18.

The power flow is shown schematically in FIG. 4 where the power P_(IN)comes into the primary (transmitting) coil 12 from the power supply.Energy is stored in the primary coil 12 in the form of an oscillatingelectric and magnetic field. This energy P_(k) may be either transmittedto the secondary (receiving) coil 16 via the channel 40, or lost aspresented by P_(Γ1).

The received power at the secondary (receiving) coil 16 also may beeither transmitted (Pw) to the power consuming device, or lost as shownby P_(Γ2).

The mechanisms for power loss are attributed either to the resistivelosses in the coil wire, losses in the dielectric material that may beused in the construction of the coils and/or the capacitors, as well asto the losses due to radiated energy. It is clear that by lowering thelosses in the system, a higher energy level may be available for thesecondary coil 16, and a higher overall efficiency of the system 10 isattained.

In order to reduce the losses in the present system, multiple approacheshave been considered and implemented, including:

-   -   A. Use of superconducting material in the construction of the        components,    -   B. Minimizing or avoiding the use of dielectric materials in the        system components, and    -   C. Lowering the operating frequency (to below 1 MHz).

With respect to lowering the operating frequency, it is to be taken inconsideration that the operating frequency cannot be lowered excessivelysince this approach will lower the amount of power that may betransmitted. Therefore, a balance has to be maintained between loweringthe operating frequency for increasing the efficiency with which thepower may be transmitted and keeping the satisfactory amount oftransmitted power.

As presented in FIGS. 5 and 6, the transmitting and receiving oscillatorcoils 12 and 16 are contoured as compact flat multiturn coils. Asopposed to the helical geometry of the oscillators used in prior systemsshown in FIG. 1 which was used to reduce the self capacitance of thecoil in order to target a particular frequency (10 MHz) which wasoptimal with the regular conductors being used, the present system useslower frequencies (below 1 MHz). Therefore a compact flat spiral contouris used which is advantageous in occupying a lesser volume which isimportant in the case of using a superconducting coil in the presentsystem since it facilitates enclosing the coil in the cryogenicequipment 42, 43 schematically shown in FIGS. 2 and 3.

Shown in FIG. 6, the coil of the present invention is held in close-upposition by a structure 44 which holds each turn of the coil spaced apredetermined fixed distance each from the other without introducing asignificant amount of dielectric material between the windings. Thereare several structures 44 holding the coil in close-up shape as shown inFIG. 5.

Although, it is possible to manufacture the coils 12, 16 with a thinlayer of dielectric 45 between the windings as shown in FIG. 8, it ispreferable that there is no dielectric material used between the turnsof the coils in order to diminish or eliminate dielectric losses in thecoil.

The principle of the oscillator design, as well as the detailedprinciple of the design of the capacitors 36, 38 will be presented infollowing paragraphs.

Referring to FIGS. 7-9 an air-gap capacitor 36, 38 is constructed fromtwo coaxial cylinders 50 and 52 having different radii (a<b). Shown inFIG. 8 is the connection of the ends of the coil 12 or 16 to thecylindrical capacitor 36, 38 where one end 28, 32 of the coil 12, 16,respectively, is connected to the outer cylinder 52, while another end30, 34 of the coil 12, 16, respectively, is connected to the innercylinder 50. The inner and outer cylinders 50 and 52 of the capacitor 36or 38 each corresponds to a respective electrode of the capacitor.

In order that the system 10 be functional, the capacitor 36, 38 does nothave to be superconducting or even dielectric-less. However, currentflowing in and out of the capacitor will suffer from ohmic (resistive)losses if it is fabricated from a regular conductor material, andlikewise there will be dielectric losses present if a dielectric isused. FIG. 9 shows a section of the air gap 62 filled with a dielectric47.

The capacitor design shown with two cylindrical electrodes is anexemplary embodiment. Any geometry may be used for the capacitor inquestion. The advantage of the concentric cylinders, however, is thatsuch permits the use of a magnetic field to act as a dielectric medium.

The purpose of the dielectric in a capacitor is to allow electrodes ofthe capacitor to be placed closer together in order to raise thecapacitance. Without the dielectric, the electrons penetrate the gapbetween the electrodes and cause shorting in the capacitors. However,molecules of the dielectric material respond to the changing electricalfield by trying to align with it, thus resulting in their oscillation.This effect causes an internal friction that dissipates energy causing adielectric loss. By using a magnetic field, instead of the dielectric,the electrons can be partially stopped or blocked from crossing theelectrode gap, thus allowing the electrodes to be closer togetherwithout the risk of shorting. At the same time, since there is nomaterial present, the usual dielectric losses are not seen.

The superconducting capacitor may be made of the same material as thewire of the primary and secondary coils in the system, for instance,BCCO or YBCO, or of a Type I superconductor, or a room temperaturesuperconductor. However, as opposed to the ribbon wire of the coils, theelectrodes in the capacitor are formed from solid pieces of thesuperconducting material.

Alternatively, as shown in FIG. 9, the capacitor also may be constructedfrom several concentric electrodes. In the example shown in FIG. 9,there are four concentric cylinders 52, 58, 60 and 63 in total connectedalternately to the ends 28, 32, and 30, 34 of the oscillators 12, 16,respectively, thus increasing the overall capacitance. The outercylinder 52 encircles one of the inner cylinder electrodes 58 which inturn encircles the electrode 60, etc. Any number of the concentricallydisposed cylinders may be used for the capacitor design.

The theoretical and design principles of the oscillating structures aswell as the capacitors in the subject system 10 will be presentedfurther herein.

The coupled-mode theory (CMT) presented in Haus, et al., “Waves andFields in Optoelectronics”, Prentice-Hall, N.J. (1984) is a convenientframework to analyze performance of the subject system 10. For smallloss levels, the formalism provides for a more physical understanding ofthe relevant processes.

Using a resonant circuit shown in FIG. 3, a complex amplitudecorresponding to the instantaneous power is defined as

$\begin{matrix}{{a = {{\sqrt{\frac{C}{2}}v} + {j\sqrt{\frac{L}{2}}i}}}\mspace{14mu} {W = {{a*a} = {{\frac{C}{2}V_{\max}^{2}} = {\frac{L}{2}I_{\max}^{2}}}}}} & ( {{Eq}.\mspace{14mu} 1} )\end{matrix}$

where C, L are the capacitance and inductance, V_(max), I_(max) are thepeak voltage and current, v, i are the instantaneous voltage and currentand W is the total energy contained within the circuit. With thisdefinition, losses that are small with respect to the recirculated powercan be introduced as

$\begin{matrix}{\frac{a_{1}}{t} = {{{j\omega}_{0}a_{1}} - {\Gamma_{1}a_{1}} + {{j\kappa}_{12}a_{2}}}} & ( {{Eq}.\mspace{14mu} 2} )\end{matrix}$

where Γ₁a₁ relates to an unrecoverable drain of power to the environmentand κ₁₂a₂ is an exchange of power with a second resonant device withcomplex amplitude a_(2 .)

It may be shown by energy conservation that under this definition thecoupling coefficients must be equal (κ₁₂=κ₂₁=κ) and it will be assumedthroughout that the oscillators 12, 16 are identical (Γ₁=Γ₂=Γ. Thefigure of merit (FOM) of such a configuration is given by κ/Γ which maybe seen as the rate of power coupling divided by the rate of powerdissipation. The regime of interest where this quantity is much greaterthan one is referred to as “strong coupling”.

As stated earlier, in the system for power transfer, the two sources ofdissipative losses are ohmic and radiative. At radio frequencies, thecurrent travels on the outer surface of the conductor (skin effect) andthe characteristic skin depth is given by

$\begin{matrix}{\delta = \sqrt{\frac{2\rho}{\omega\mu}}} & ( {{Eq}.\mspace{14mu} 3} )\end{matrix}$

where ρ is the resistivity, ω is the frequency, and μ is the materialpermeability.

For small values of δ compared to wire radius, the resistance istherefore

$\begin{matrix}{R_{ohm} = {{\rho \frac{l}{A}} = {{\frac{4\pi^{2}{RN}}{w}\sqrt{\rho \overset{\_}{f}}} = {c_{ohm}\sqrt{\overset{\_}{f}}}}}} & ( {{Eq}.\mspace{14mu} 4} )\end{matrix}$

where R, N and w are the coil radius, number of turns in the coil andwire width respectively, and f=f/(10 MHz). The wire of the oscillatingcoils 12, 16, although other implementations are envisioned, as anexample, is assumed to be a ribbon with a width (w) much greater thanits thickness. The radiative losses are given in the quasi-static limitas—presented in [C. Balanis, et al., “Antenna Theory: Analysis andDesign”, Wiley, N.J., (2005)]:

$\begin{matrix}{{R_{rad} \approx {\frac{8\pi^{3}}{3}\sqrt{\frac{\mu_{0}}{ɛ_{0}}}( \frac{NA}{\lambda^{2}} )^{2}}} = {c_{rad}{\overset{\_}{f}}^{4}}} & ( {{Eq}.\mspace{14mu} 5} )\end{matrix}$

From Eqs. 1 and 2, the rate of energy dissipation is

$\begin{matrix}{\frac{W_{loss}}{t} = {{{- 2}\; \Gamma \; W} = {{{- 2}{\Gamma ( {\frac{L}{2}I_{\max}^{2}} )}} = {{- \frac{I_{\max}^{2}}{2}}R_{diss}}}}} & ( {{Eq}.\mspace{14mu} 6} )\end{matrix}$

where the last equality is simply an identification of the usual ohmicdissipation as a function of the RMS (Root Mean Square) current, whichis I_(max)/√{square root over (2)}.

From this it may be seen that the loss coefficient can be related to theinductance and dissipation of the coil by

$\begin{matrix}{\Gamma = {\frac{R_{diss}}{2L} = {\frac{1}{2L}( {R_{ohm} + R_{rad}} )}}} & ( {{Eq}.\mspace{14mu} 7} )\end{matrix}$

The coupling coefficient is defined in terms of the mutual inductance by

$\begin{matrix}{\kappa = \frac{\omega \; M}{2L}} & ( {{Eq}.\mspace{14mu} 8} )\end{matrix}$

where again the coils are assumed identical with inductance L.

In the quasi-static limit and at large distances D>R the magnetic fluxdensity at the secondary coil 16 as a result of the primary coil 12 hasthe form of a dipole

$\begin{matrix}{B = {{\frac{\mu_{0}}{4\pi}\frac{NiA}{D^{3}}\sqrt{1 + {3\; \sin^{2}\theta}}} \approx {\frac{\mu_{0}}{2}\frac{{NiR}^{2}}{D^{3}}}}} & ( {{Eq}.\mspace{14mu} 9} )\end{matrix}$

where coaxial orientation of the coils has been assumed in the lastapproximation. The mutual inductance is then found from the flux throughthe N linkages of the secondary coil 16 as

$\begin{matrix}{M = {{N\frac{\partial\Phi}{\partial i}} \approx {{NA}\frac{\partial B}{\partial i}} \approx {\frac{\pi}{2}N^{2}\mu_{0}\frac{R^{4}}{D^{3}}}}} & ( {{Eq}.\mspace{14mu} 10} )\end{matrix}$

The ratio of coupled to dissipated power follows

$\begin{matrix}{\frac{\kappa}{\Gamma} = {\frac{\omega \; M}{R_{diss}} = {4\pi^{3}\frac{N^{2}}{c_{rad}}\frac{R^{4}}{D^{3}}\frac{\overset{\_}{f}}{{\frac{c_{ohm}}{c_{rad}}\sqrt{\overset{\_}{f}}} + {\overset{\_}{f}}^{4}}}}} & ( {{Eq}.\mspace{14mu} 11} )\end{matrix}$

and from the definition of c_(rad), the FOM becomes

$\begin{matrix}{\frac{\kappa}{\Gamma} \approx {\frac{326}{D^{3}}\frac{\overset{\_}{f}}{{\frac{c_{ohm}}{c_{rad}}\sqrt{\overset{\_}{f}}} + {\overset{\_}{f}}^{4}}}} & ( {{Eq}.\mspace{14mu} 12} )\end{matrix}$

where D is in meters.

It may be seen that the FOM goes to zero at low frequencies as a resultof slowly decreasing ohmic losses, as well as at high frequencies as aresult of rapidly increasing radiative losses. The frequency dependenceof the ohmic losses shown in Eq. 12 does not actually hold at very lowfrequencies where it approaches a constant value, however this does notchange the result that follows. The FOM therefore has a maximum, and itsvalue and corresponding frequency are easily found from differentiatingEq. 12

$\begin{matrix}{{\frac{\kappa}{\Gamma}_{\max}{\approx {\frac{216}{D^{3}}( \frac{c_{rad}}{c_{ohm}} )^{\frac{6}{7}}{\overset{\_}{f}}_{opt}}}} = ( {\frac{1}{6}\frac{c_{ohm}}{c_{rad}}} )^{\frac{2}{7}}} & ( {{Eq}.\mspace{14mu} 13} )\end{matrix}$

The loss mode ratio governing the optimal frequency is related to thecoil design parameters by

$\begin{matrix}{\frac{c_{ohm}}{c_{rad}} \approx {52\frac{\sqrt{\rho}}{{wNR}^{3}}}} & ( {{Eq}.\mspace{14mu} 14} )\end{matrix}$

For a copper coil used in the prior art system with parameters R=30 cm,w=4 mm, N=1 is seen to have an optimal frequency of 10.6 MHz ( f=1.06)and a corresponding coupling ratio (FOM) equal to 117. Eqs. 13 and 14provide a method of designing a set of coils for a particular frequencyin such a way as to maximize the coupling with minimum loss.

An immediate consequence of the result of the analysis presented suprais that a reduction in the resistivity of the oscillator provides for anincrease in the efficiency of the coupling. By eliminating the resistivelosses of the primary and secondary coils 12, 16, there is freedom todrive the frequency to lower values, reducing the radiative losses aswell. FIG. 10 shows the frequency dependent portion of the FOM (Eq. 12)for three different loss mode ratios. As expected, the optimal value ofthe frequency is tending toward lower frequencies in the limit that theresistive losses are zero.

It is instructive to adopt the definition given in [A. Kurs, et al.,“Wireless Power Trans. Via Strongly Coupled Mag. Resonances”, Science,317 (2007)] for the efficiency of the power transfer, given as

$\begin{matrix}{\eta = \frac{\Gamma_{W}{a_{2}}^{2}}{{\Gamma {a_{1}}^{2}} + {( {\Gamma + \Gamma_{W}} ){a_{2}}^{2}}}} & ( {{Eq}.\mspace{14mu} 15} )\end{matrix}$

where 2Γ|a₁|² and 2Γ|a₂|² are the rates of power dissipation in eachoscillator coil as discussed in the previous paragraphs, and 2Γ_(W)|a₂|²is the rate of power transferred into a load coupled to the secondarycoil 16.

A similar loss may be attributed to the amplifier driving the primarycoil 12, but the efficiency will be defined as relative to the outputpower of the amplifier. In steady-state, the power that is coupled tothe secondary coil 16 must equal the total power consumed by bothdissipation and the load draw. This results in a relationship betweenthe energy content of each resonator

κ² |a ₁|²=(Γ+Γ_(W))² |a ₂|²  (Eq. 16)

and allows the efficiency to be expressed solely in terms of loss andcoupling parameters

$\begin{matrix}{\eta = \frac{\frac{\Gamma_{W}}{\Gamma}( \frac{\kappa}{\Gamma} )^{2}}{( {1 + \frac{\Gamma_{W}}{\Gamma}} )^{2} + {( {1 + \frac{\Gamma_{W}}{\Gamma}} )( \frac{\kappa}{\Gamma} )^{2}}}} & ( {{Eq}.\mspace{14mu} 17} )\end{matrix}$

Differentiating this expression with respect to Γ_(W)/Γ, the maximumefficiency occurs when

Γ_(W) ²−Γ²=κ²  (Eq. 18)

and at this condition, the efficiency may be expressed as

$\begin{matrix}{\eta_{\max} = {\frac{\Gamma_{W} - \Gamma}{\Gamma_{W} + \Gamma} = \frac{\sqrt{1 + ( \frac{\kappa}{\Gamma} )^{2}} - 1}{\sqrt{1 + ( \frac{\kappa}{\Gamma} )^{2}} + 1}}} & ( {{Eq}.\mspace{14mu} 19} )\end{matrix}$

While efficiency is certainly a driver of the subject system design, theactual power level that can be transferred to a load 18 on the receivingend is of a great importance as well. This power is related to therecirculating energy of the coil and using Eq. 16 may be expressed as

$\begin{matrix}\begin{matrix}{P_{W} = {2\Gamma_{W}{a_{2}}^{2}}} \\{= {\frac{2\Gamma_{W}\kappa^{2}}{( {\Gamma + \Gamma_{W}} )^{2}}{a_{1}}^{2}}} \\{= {2{\Gamma ( \frac{\kappa}{\Gamma} )}^{2}( \frac{\Gamma_{W}}{\Gamma} )( {1 + \frac{\Gamma_{W}}{\Gamma}} )^{- 2}( {\frac{1}{2}{LI}^{2}} )}}\end{matrix} & ( {{Eq}.\mspace{14mu} 20} )\end{matrix}$

This expression may also be differentiated with respect to Γ_(W)/Γ tofind the ratio of load to dissipation that maximizes the power. Thisresults in Γ_(W)/Γ=1, and the maximum power to the load is therefore

$\begin{matrix}{P_{W} = { {\frac{\Gamma}{4}( \frac{\kappa}{\Gamma} )^{2}( {LI}^{2} )}\Rightarrow\eta  = {\frac{1}{2}( {{2( \frac{\kappa}{\Gamma} )^{- 2}} + 1} )^{- 1}}}} & ( {{Eq}.\mspace{14mu} 21} )\end{matrix}$

showing that for maximum power transfer, the dissipation should be at aminimum and the energy content and coupling should be at a maximum. Alsoshown is the resulting efficiency at the condition of maximum powertransfer, and it may be seen that the efficiency approaches a maximumvalue of 50% as the FOM increases.

To estimate the power and efficiency at a nominal distance of 100meters, Eq. 12 is used in the superconducting limit

$\begin{matrix}{\frac{\kappa}{\Gamma} \approx \frac{326}{( {\overset{\_}{f}D} )^{3}}} & ( {{Eq}.\mspace{14mu} 22} )\end{matrix}$

and at 200 kHz, f=0.02, so that κ/Γ≈41 and η_(max)≈95%.

To estimate the power delivered, the product of ΓL is given by Eqs. 5and 7 as ΓL=½C_(Rad) f ⁴, again with no ohmic losses.

Assuming a coil design having 100 turns of wire and a radius of 0.5meters, c_(Rad)=237. At maximum efficiency

${\frac{\Gamma_{w}}{\Gamma} = 0.026},$

and if I_(max)=100 amps then the power delivered at 100 meters is 7.8Watts. However, if the maximum power coupling is desired, Γ_(W) ismatched to Γ, and from Eq. 21 the efficiency is ˜50%. The powerdelivered at 100 meters in this case is 80 Watts. From Eqs. 19 and 21,and from the performance numbers just discussed, it is seen that thereis a conflict between maximizing efficiency and maximizing powerdelivered.

Consider the case when maximizing the efficiency is the goal. Eq. 19shows that the efficiency is not appreciably affected until κ/Γ dropsbelow a value of around 3, corresponding to a frequency-distance product( fD) approaching ˜5. At this point the maximum efficiency drops to 50%and at 200 kHz this corresponds to a distance of 250 meters. Reducingthe frequency by a factor of 10 will extend this 50% value of maximumefficiency out to 2.5 km. The relationship between maximum efficiencyand fD in the limit of no ohmic dissipation is shown in FIG. 11.

The results discussed above are for optimal efficiency, where the powerdrawn by the load is related to the radiative losses and couplingcoefficient by Eq.

18. This condition can be maintained by actively varying the load toshunt excess power into onboard storage or extract it from storage whenneeded. It is of interest to investigate how the efficiency changes as aresult of off-nominal operations to assess how closely such a systemwould have to track power usage.

In the strong coupling regime, the maximum efficiency occurs whenΓ_(W)/κ≈1, but from Eq. 17, the efficiency can be found that resultsover a continuum of load to coupling ratios, spanning either side of theoptimal value.

In FIG. 12, the variation of efficiency with Γ_(W)/κ for several valuesof κ/Γ is shown. It may be seen that the sensitivity of the efficiencyto a mismatch in power goes down as the coupling strength increases. Infact, the half-max value of the efficiency occurs whereΓ_(w)/κ=(κ/Γ)^(±1). The FOM may therefore also be interpreted as theeffective ‘bandwidth’ for efficient power coupling. Thus, operating inthe strong coupling regime means that closely matching the load to anoptimal value is not critical, and in fact a considerably large mismatchmay only reduce the efficiency by an acceptably small amount.

Oscillator Design Considerations

In the previous paragraphs, a frequency of 200 kHz was selected for usewith a superconducting oscillator 12, 16 to improve the efficiency overwhat can be achieved using non-superconducting materials. As opposed tothe design of “WiTricity” where a regular conducting wire was used inorder to create a coil that has a natural resonance in the 10 MHz rangeto provide maximum efficiency, helical coils having large torsion wereused as a way of reducing their self-capacitance. Since lowerfrequencies (below 1 MHz, and, for example, preferably at or below 200KHz) are of interest in the present system, more compact coils areconsidered for the oscillators 12, 16.

The subject oscillators 12, 16 are formed from a superconducting wire,for example, one that is commercially available and is formed in theshape of ribbon that can be conveniently wound into a flat spiral 12,16. An example of such wire is Bismuth Strontium Calcium Copper Oxide(BSCCO). BSCCO (“bisko”) is a family of High Temperature Superconductors(HTS), having a critical temperature of around 110 K. As such, they canbe easily cooled using liquid nitrogen (77 K at 1 atmosphere), or byusing a thermocontroller 42 for controlling a plurality of cryo-coolers43, as schematically shown in FIGS. 2 and 3. BSCCO is a high temperaturesuperconductor which does not contain a rare earth element, and isformed into wires and offered commercially. Also, Type I superconductormaterials, as well as room temperature superconductors, may be used as amaterial of the spirals 12, 16. In one embodiment, a dielectric 45, suchas 2 mil Kapton tape, may be present between the windings of the coils12, 16, as shown in FIG. 8.

The inductance for such a winding is given by {E. Rosa, “The Self andMutual Induct. Of Linear Conductors”, B. of the Bureau of Standards, 4,2 (1908)].

$\begin{matrix}{L \approx \frac{4\mu_{0}R^{2}N^{2}}{R + {1.4h}} \approx {4\mu_{0}{RN}^{2}\mspace{20mu} ( {{{for}\mspace{14mu} R} > h} )}} & ( {{Eq}.\mspace{14mu} 23} )\end{matrix}$

where h (hereafter assumed negligible) is the difference between theinner and outer radii of the windings and the other parameters are aspreviously defined.

The issue of “self-capacitance” of the oscillators 12, 16 is morecomplicated. The flat spiral can be thought of as a long parallel platecapacitor, wrapped around onto itself. The capacitance per unit lengtharound the spiral is a constant, however the voltage distribution acrossthe capacitor is not constant from one end to the other, resulting in avariation in the energy distribution. With the ends 38 of the spiral 30unconnected, the fundamental frequency of the coil will correspond to ahalf wavelength standing wave across the entire coil, such that thecurrent in the coil goes to zero at the ends. The distribution ofcurrent, charge and potential around the spiral can be given by

$\begin{matrix}{{{I( {x,t} )} = {I_{\max}{\sin ( {\pi \; {x/l}} )}\cos \; \omega \; t}}{{q( {x,t} )} = {{- q_{\max}}{\cos ( {\pi \; {x/l}} )}\sin \; \omega \; t}}{{V( {x,t} )} = {\frac{V_{\max}}{2}{\cos ( {\pi \; {x/l}} )}\sin \; \omega \; t}}} & ( {{Eq}.\mspace{14mu} 24} )\end{matrix}$

where the time phase of the current has been arbitrarily chosen, and theresulting functional forms of charge and potential satisfy chargeconservation and Poisson's equation around the spiral. V_(max) has beendefined as the potential across the entire coil (end to end) and will beused to define the capacitance.

The energy per unit volume of insulator between the windings of the coilat any given location around the spiral is given by

$\begin{matrix}{W = {{\frac{1}{2}ɛ_{r}ɛ_{0}E^{2}} = {\frac{1}{2}ɛ_{r}{ɛ_{0}\lbrack \frac{{V(x)} - {V( {x + {2\pi \; R}} )}}{d} \rbrack}^{2}}}} & {( {{Eq}.\mspace{14mu} 25} )\mspace{11mu}}\end{matrix}$

where the voltage difference is taken between any point on the spiral12, 16 and the nearest point one turn farther along the spiral, directlyacross the dielectric between the windings. Multiplying by thecross-section of the dielectric (wd), and integrating over the length ofthe spiral (1≈2 πRN), the maximum total energy stored in the electricfield is given as

$\begin{matrix}{{\frac{1}{2}{CV}_{\max}^{2}} = {\frac{1}{2}( \frac{ɛ_{r\;}ɛ_{0}\pi^{3}{Rw}}{4{Nd}} )V_{\max}^{2}}} & ( {{Eq}.\mspace{14mu} 26} )\end{matrix}$

where N>1 has been assumed. Combining the results of Eqs. 23 and 26, thenatural frequency of the flat spiral coil is given by

$\begin{matrix}{\overset{\_}{f} \approx {\frac{8.6}{R}\sqrt{\frac{d}{ɛ_{r}{Nw}}}}} & ( {{Eq}.\mspace{14mu} 27} )\end{matrix}$

A coil radius of R=0.5 m may be assumed along with the aforementioned200 kHz target frequency. The HTS wire mentioned above may be 4 mm wide,so using 2 mil Kapton with a dielectric constant of 4.0 in this baselinedesign results in a natural frequency of 96 kHz.

The required strength of the dielectric can be found by equating theenergy contained in the magnetic field at peak current to the energystored in the electric field at zero current, given by Eq. 26. The peakcurrent (I_(max)) is related to the RMS current over ½ wavelength(denoted RMS/2) by a factor of ½, resulting in the relation

$\begin{matrix}{{\frac{1}{2}{LI}_{{RMS}/2}^{2}} = {{\frac{1}{8}{LI}_{\max}^{2}} = {\frac{1}{2}( \frac{ɛ_{r}ɛ_{0}\pi^{3}{Rw}}{4{Nd}} )V_{\max}^{2}}}} & ( {{Eq}.\mspace{14mu} 28} )\end{matrix}$

and consequently

$\begin{matrix}{V_{\max} = {{( \frac{4\mu_{0}N^{3}d}{ɛ_{r}ɛ_{0}\pi^{3}w} )^{\frac{1}{2}}I_{\max}} \approx {135\sqrt{\frac{N^{3}d}{ɛ_{r}w}}I_{\max}}}} & ( {{Eq}.\mspace{14mu} 29} )\end{matrix}$

From Eq. 24, the maximum potential difference across the dielectric willoccur near the center of the coil length where the gradient of thepotential around the coil is the largest. Multiplying the maximumgradient by the length of a single turn, the largest potential seenacross the dielectric is given by

$\begin{matrix}{{\Delta \; V_{\max}} = {{( \frac{\mu_{0}{Nd}}{ɛ_{r}ɛ_{0}\pi \; w} )^{\frac{1}{2}}I_{\max}} \approx {213\sqrt{\frac{Nd}{ɛ_{r}w}}I_{\max}}}} & ( {{Eq}.\mspace{14mu} 30} )\end{matrix}$

Using the baseline numbers, the maximum electric field seen across thedielectric will be 262 V/mm, which is well below the dielectric strengthof Kapton (197 kV/mm).

A problem associated with this design results from dissipative losses inthe dielectric. The dissipation factor, also known as the loss tangentunder the relation DF=tan δ, represents the ratio of resistive powerloss to reactive power in the capacitor. Under the current formalism,the dissipation factor is introduced in the same manner as the previousresistive loss term

$\begin{matrix}{\Gamma_{DF} = {\frac{R_{DF}}{2\; L} = {\frac{L\; \omega \; \tan \; \delta}{2\; L} = {\frac{\omega}{2}\tan \; \delta}}}} & ( {{Eq}.\mspace{14mu} 31} )\end{matrix}$

where the matching of the reactance of the coil 12, 16 and capacitor 36,38 at resonance has been used. This result is somewhat problematic forthe efficiency, since the dissipation in the dielectric scalesidentically to the power that can be coupled. It is therefore of utmostimportance either to eliminate the dielectric whatsoever from the coils12, 16 and the capacitors 36, 38, or to use dielectrics with thesmallest possible dissipation factor.The effect of DF (dissipation factor) can be inserted directly into Eq.12, resulting in

$\begin{matrix}{\frac{\kappa}{\Gamma} \approx {\frac{326}{D^{3}}\frac{1}{\frac{c_{DF}}{c_{rad}} + {\overset{\_}{f}}^{3}}}} & ( {{Eq}.\mspace{14mu} 32} )\end{matrix}$

where elimination of the resistive losses in the coil has still beenassumed. The ratio of the loss coefficients is then

$\begin{matrix}{\frac{c_{DF}}{c_{rad}} = {{830\frac{\tan \; \delta}{R^{2}}} \approx {830\frac{\delta}{R^{2}}}}} & ( {{Eq}.\mspace{14mu} 33} )\end{matrix}$

and the need for a low dissipation factor becomes quite evident. At lowfrequencies, the dissipation in the dielectric will dominate theradiative losses, just as was the case for the ohmic losses previously.

To get a feel for this limitation, consider the use of fused quartz as adielectric, which has a relatively low dissipation factor (˜10⁻⁴). Forthe baseline design of R=0.5 m, the ratio in Eq. 33 is approximatelyequal to 0.33. The dissipation due to the dielectric would then becomparable to the radiation losses at a frequency of f=≈0.69 or about 7MHz. Evidently, a comparable loss level to what existed before thesuperconducting wire was introduced has returned.

To return to the efficiency offered by reducing the frequency to below 1MHz, for example to about or below ˜200 kHz, the ratio of dissipativeloss to radiative loss given in Eq. 33 would have to be reduced by afactor of at least 10⁻⁵. This does not appear to be a viable approachunless the dielectric is removed altogether as shown in the presentedembodiment of FIGS. 5-6. Even the presence of a cryogen such as liquidnitrogen to cool the superconducting wire will result in a loss tangentof approximately 5(10⁻⁵), so designs apparently must avoid the use ofany dielectric in the inter-electrode space.

Superconducting Capacitor Design

Once the ohmic and dielectric losses are removed from the coil 12, 16,the resulting natural frequency may not be at the level desired for thesystem operation. Further reduction in frequency may be made possible byadding more capacitance 36, 38 externally to the coil 30, as shown inFIGS. 2, 3 and 8. The use of capacitors without dielectrics is desiredin order to retain the high Q-values of the superconducting oscillator12, 16. The limitation of the dielectric-free capacitor 36, 38 is thelow dielectric strength of the remnant gas in the gap. An approach takenherein with the goal to increase the breakdown voltage of the capacitor36, 38 without introducing dielectric loss is to inhibit the avalancheionization of the gas medium by applying a magnetic field.

As shown in FIGS. 7 and 8, an air-gap capacitor 36, 38 is constructedfrom two coaxial cylinders 50, 52 of radii a and b (with a<b),respectively. An axial magnetic flux density B _(z) is applied, eitherby placing permanent magnets at the end of the cylinders, or by inducingan azimuthal current in the outer cylinder. In FIG. 7, I_(C) is thecircuit current between the capacitor 36, 38 and the coil 12, 16,respectively, and I_(B) represents a steady current that could producethe desired magnetic field.

The electric field in the capacitor 36, 38 is purely radial, but itsmagnitude changes in time cyclically over a period 1/(10 f) μsec. As theelectric field at the surface of each cylinder 50, 52 increases,electrons will leave the surface until an avalanche occurs near thedielectric strength of the air gap. In the presence of a uniform axialmagnetic field, the motion of electrons that leave the surface of eithercylinder can be idealized to three primary components—cyclotron, E×Bdrift and polarization drift. These are given respectively as:

$\begin{matrix}{{{\overset{harpoonup}{v}}_{c} = {\sqrt{2{mK}}{\hat{e}}_{\bot}}}{{\overset{harpoonup}{v}}_{EB} = {\frac{E_{r}}{B_{z}}{\hat{e}}_{\theta}}}{{\overset{harpoonup}{v}}_{p} = {\frac{1}{\Omega}\frac{E_{r}}{B_{z}}{\hat{e}}_{r}}}} & ( {{Eq}.\mspace{14mu} 35} )\end{matrix}$

where the first notion describes perpendicular motion around themagnetic field lines, the second notion is azimuthal motion around theinner electrode cylinder, and the third notion corresponds to the radialmotion across the air gap. In these expressions, K is the kinetic energyof the electron, and Ω is the electron gyro frequency.

If the cyclotron radius is small compared to the air gap 62, then themotion will be a superposition of these small gyrations with a bulkspiral motion whose direction depends on the changing electric field. Bylimiting the otherwise unimpeded radial motion of free electrons to thatof the polarization drift velocity across the field lines, theexpectation is that breakdown in the air gap 62 can be suppressed.

From Gauss' law, the radial electric field is found as a function ofradial position and substituted into the polarization drift expressionto yield

$\begin{matrix}{v_{p} = {r = {{\frac{1}{\Omega}\frac{E_{r}}{B_{z}}} = \frac{m_{e}q}{{eB}_{z}^{2}C\; {\ln ( {b/a} )}r}}}} & ( {{Eq}.\mspace{14mu} 36} )\end{matrix}$

Separating variables and integrating produces

$\begin{matrix}{\frac{r_{2}}{r_{1}} = \lbrack {1 + \frac{2{m_{e}( {q_{2} - q_{1}} )}}{{eB}_{z}^{2}C\; {\ln ( {b/a} )}r_{1}^{2}}} \rbrack^{1/2}} & ( {{Eq}.\mspace{14mu} 37} )\end{matrix}$

giving the ratio of initial to final radial position of the electronover a change in the inner conductor charge state. FIG. 13 shows thevariation in charge state of the inner (Qinner) and outer (Qouter)conductors over a full cycle, indicating the notional regions (regions1, 2) where significant electron release would typically occur as aresult of the electric field at the conductor surface. Note that theelectron emission from the inner electrode 52 will have a lowerthreshold due to a smaller radius of curvature and higher electricfield. Also indicated by shading are the regions where free electronswill drift outward (region A) or inward (region B) across the magneticfield lines due to the polarization drift.

While dielectric breakdown would normally initiate as the chargemagnitude on the inner conductor passes a critical negative value(region 2), the polarization drift forces these electrons back towardthe inner electrode surface (region B).

It is not until the charge reaches its peak and the polarization driftchanges direction (region A—right) that the charges may start to migrateoutward. Migration continues until the polarization drift again changesdirection (region B) and the charges begin to migrate back toward theinner electrode. By the time the outer electrode 52 reaches its peaknegative value, these charges have been returned to the inner electrode50 where they started, and the cycle repeats. The same process occursfor electrons emitted from the outer electrode over the other half ofthe cycle.

The capacitor size and magnetic field strength are to be chosen toensure that the electrons are turned around prior to reaching theopposite electrode. The full change in charge state of the capacitor 36,38 over the time it takes for the electrons to cross the gap is2q_(max)=2I_(max)/ω. The limiting ratio of electrode radii is then foundusing Eq. 37

$\begin{matrix}{\frac{b}{a} > \lbrack {1 + \frac{4m_{e\;}I_{\max}}{e\; \omega \; B_{z}^{2}C\; {\ln ( {b/a} )}a^{2}}} \rbrack^{1/2} \approx {\frac{N}{113B_{z}a_{cm}}( \frac{{RI}_{\max}\overset{\_}{f}}{\ln ( {b/a} )} )^{1/2}}} & ( {{Eq}.\mspace{14mu} 38} )\end{matrix}$

where a_(cm) is the inner cylinder 50 radius in centimeters, and theother terms are as previously defined. To find the capacitor length, aset of concentric cylinders is assumed with an impedance matched to thatof the coil 12, 16 at resonance.

This assumes the capacitance of the coil 12, 16 is negligible, but itcould be included in the calculation at this point. Ignoring the coilcapacitance results in

$\begin{matrix}\begin{matrix}{C = {\frac{2{\pi l}\; ɛ_{0}}{\ln ( {b/a} )} =  \frac{1}{\omega^{2}L}\Rightarrow l_{cm} }} \\{= {{100\frac{\ln ( {b/a} )}{2{\pi ɛ}_{0}\omega^{2}L}} \approx {91\frac{\ln ( {b/a} )}{{\overset{\_}{f}}^{2}N^{2}R}}}}\end{matrix} & ( {{Eq}.\mspace{14mu} 39} )\end{matrix}$

where l_(cm) is the capacitor length in centimeters. Substituting Eq. 38into Eq. 39 then implies

$\begin{matrix}{{\pi \; b_{cm}^{2}l_{cm}} \approx {\frac{I_{\max}}{45\overset{\_}{f}B_{z}^{2}}\mspace{14mu} {cm}^{3}}} & ( {{Eq}.\mspace{14mu} 40} )\end{matrix}$

showing that the volume of the cylindrical capacitor is driven by thechoice of maximum current, frequency and magnetic field. The mostcompact geometry is when l_(cm)≈2b_(cm), and the capacitor 36, 38 fitswithin a cube. For the baseline case and assuming an applied magneticflux density of 0.3 Tesla, the volume given by Eq. 40 is 216 cm³. Thisresults in a capacitor length of 12 cm, an outer electrode radius of 6.0cm and an inner electrode radius of 4.6 cm.

Returning to FIG. 2, in order to boost the range of power transfer,passive repeaters 70 are envisioned in the subject system 10.

The passive repeaters 70 are the intermediate coils which resonate inphase with the primary oscillator 12, to receive the power therefrom,and in phase with the secondary resonator 16 to further transfer powerto the secondary oscillator 16.

Attenuation and Environmental Coupling

There are a variety of ways in which the transmitting 12 and receiving16 coils can interact with the environment, potentially resulting in ashift in the frequency, attenuation of the delivered power or reductionof overall efficiency. The degree to which each of these would occur ishighly specific to the properties and distribution of materials withinthe reach of the magnetic field. A general overview of the effects andtheir insertion into the current formalism may be discussed.

The frequency shift results from a change of the effective inductance ofeither of the coils, resulting from the presence of a material withmagnetic permeability above unity, similar to placing an iron corewithin a solenoid. The larger the volume of material present, and thecloser it is to the coil, the greater the shift in frequency that willresult. The effect of the presence of the material on the inductance canbe estimated as follows. The reactive power within the coil is given byP_(R)=LωI_(max) ², which can also be written as

$\begin{matrix}{P_{R} = {{2\; {\omega ( {\frac{1}{2}{LI}_{\max}^{2}} )}} = {2\omega {\int_{vol}^{\;}{\frac{1}{2\mu_{0}}B_{\max}^{2}{V}}}}}} & ( {{Eq}.\mspace{14mu} 41} )\end{matrix}$

In a region with a material that has a permeability greater than unity,the ratio of B_(max) with the material present to that without thematerial present is (1+χ), where χ is the susceptibility. If themagnetic field is uniform within this region, the increase in theeffective inductance, normalized by the original value is then

$\begin{matrix}{\frac{\Delta \; L}{L} = \frac{\chi \; B_{\max}^{2}\Delta \; V}{{LI}_{\max}^{2}}} & ( {{Eq}.\mspace{14mu} 42} )\end{matrix}$

where B_(max) is evaluated locally, and Δv is the volume occupied by thematerial. A term such as this can be included for each region wheremagnetic material is present. From Eq. 9, the term I_(max) ² will divideout and only geometric dependencies will remain. The amount of frequencyshift that results is found simply by taking the differential ofexpression for the resonant frequency of the coil

$\begin{matrix}{{2\frac{\Delta \; f}{f}} = {{- \frac{\Delta \; L}{L}} - \frac{\Delta \; C}{C}}} & ( {{Eq}.\mspace{14mu} 43} )\end{matrix}$

where it can be seen that a positive increase in the inductance willresult in a drop in the natural frequency of the coil. Because theeffect on each coil will be different, depending on it location in theenvironment, the simplest solution is to compensate for this shift ateach coil independently by adjusting the capacitance until the properresonant frequency is achieved.

For any given placement of the coils in a static environment, this maybe done initially and should not need to be altered. However, it wouldbe straightforward to track the frequency and dynamically update it tocompensate for changes in the environment or for the motion of either ofthe coils through the environment. This effect is therefore not thoughtto be problematic from an operational standpoint.

Likewise, attenuation of the signal by the environment is of littleconcern. Unlike electromagnetic radiation, which can be very effectivelyattenuated by the presence of conductors, the magnetic field is muchmore difficult to shield. A particular application resulting from thisphenomenon is the ability to penetrate the depths of the ocean, eitherfor the delivery of power, desirable for recharging AutonomousUnderwater Vehicles (AUVs), or for transferring information bymodulating the signal. When magnetic field attenuation is desired, it istypically necessary to completely enshroud the item to be shielded in ahigh permeability material. The attenuation results from the‘conduction’ of the field lines around, rather than through the devicethat is to be shielded. The amount of attenuation, given as the ratio ofun-attenuated to attenuated field strength is approximately

$\begin{matrix}{A \approx {\frac{\mu}{2}\frac{\Delta}{R}}} & ( {{Eq}.\mspace{14mu} 44} )\end{matrix}$

where μ is the permeability of the shielding, Δ is the thickness of theshielding and R is the characteristic size of the shielded region.Effective attenuation without excessive mass therefore requires a highpermeability material, and the smallest possible enclosure volume. Inmost cases of interest, it is unlikely that a situation will naturallyexist to produce appreciable levels of signal attenuation.

Of greater concern is the possibility of power lost to the environment,resulting in a reduction of overall efficiency. Sources of such powerloss include dipole oscillations in paramagnetic and diamagneticmaterials, the dissipation associated with eddy currents induced inconducting materials and the hysteretic loss resulting from domainreconfiguration in ferromagnetic materials. The first of these istreated in a similar manner to the dielectric losses of Eq. 31. In fact,under the proper definition of the loss tangent, the effect would beintroduced into the formalism in an identical manner. However, theevaluation of this effective loss tangent will depend on the totalvolume and distribution of this material within the dipole field, justas with the effect on induction. If at the location of the material themagnetic field is again assumed to be spatially constant, the reactivepower per unit volume at this location is given by

$\begin{matrix}{\frac{P_{R}}{\Delta \; V} = {ɛ = {{\frac{\;}{t}( {\frac{1}{2\mu_{0}}B^{2}} )} = {\frac{BB}{\mu_{0}} = \frac{\omega \; B_{\max}^{2}}{\omega_{0}}}}}} & ( {{Eq}.\mspace{14mu} 45} )\end{matrix}$

which could also be found by performing the integration of Eq. 41 overonly the volume of magnetic material. Inserting Eq. 9 for the magneticdipole field at the location of the material, and dividing by I_(max) ²gives the effective reactance of the material (per unit volume)referenced to the recirculating coil current. This is then multiplied bythe volume of material under consideration and inserted into Eq. 31 inplace of the product Lω, along with the appropriately defined losstangent of the material for magnetic dipole oscillations.

A similar situation exists for ferromagnetic materials, however thedissipation resulting from domain hysteresis is given per unit volume as

$\begin{matrix}{\frac{P_{hys}}{\Delta \; V} \approx {2{fH}_{C}B_{rem}}} & ( {{Eq}.\mspace{14mu} 46} )\end{matrix}$

where H_(C) is the coercivity of the material (where B=0) and B_(rem) isthe remnant magnetization (where H=0). The product H_(C)B_(rem) isreferred to as the energy product and represents an approximation to thearea under the hysteresis loop, provided that the material is beingfully saturated. Unless the material is very close to one of the coils,it is unlikely that this will be the case. An approximation for the casewhen saturation has not been reached can be made with

$\begin{matrix}{\frac{P_{hys}}{\Delta \; V} \approx {2{fH}_{C}{B_{rem}( \frac{H( I_{\max} )}{H_{C}} )}^{2}}} & ( {{Eq}.\mspace{14mu} 47} )\end{matrix}$

where a linear scaling with magnetic field strength has been assumedalong the magnetic flux density axis of the hysteresis diagram, as wellas along the magnetic field strength axis. This is now a function ofI_(max) ² as before, and the expression P_(hys)/I_(max) ² is thensubstituted into Eq. 7 as a resistance term.

Finally, we consider the case of induced eddy currents in conductorsthat may be present. From a straightforward application of Faraday'sLaw, the power dissipated per unit volume from currents induced in aconductor with finite resistance scales approximately as

$\begin{matrix}{\frac{P_{eddy}}{\Delta \; V} \approx {\sigma ( {{fB}_{\max}d} )}^{2}} & ( {{Eq}.\mspace{14mu} 48} )\end{matrix}$

where σ is the conductivity of the material, and d is the characteristicsize of the region perpendicular to the local magnetic field direction.Because the loss per unit volume is seen to scale with the size of theregion, it is important to distinguish between a single continuousregion of conducting material versus a region of comparable size wheremultiple unconnected sub-domains of conducting material may exist. Asabove, this loss can be converted into a resistance, referenced to therecirculating current in the coil, and inserted into the formalism viaEq. 7.

For all of the cases except for the attenuation (shift in frequency, orthe various dissipation mechanisms) the dependence of the effect inquestion on the local magnetic field strength is quadratic. For aconstant volume of a given magnetic material, Eq. 9 shows that theimpact this material will have on the system performance scales withdistance to the center of the coil as D⁻⁶, and the impact of theseeffects rapidly diminishes with distance.

For instance, if a volume of iron placed one meter from the coil wasable to shift the frequency by 10%, at two meters the effect would bereduced to only 0.16%. Proper placement of the coil within theenvironment can therefore significantly reduce the effects that havebeen discussed. Alternatively, the coupling between the coils has thesame D⁻⁶ scaling. So, as a fraction of the power coupled, the effect ofthese materials is scale invariant. In other words, a material placedhalfway between two coils that dissipated 10% of the total coupledpower, would still dissipate 10% of the total coupled power if theseparation distance between the coils was increased by a factor of ten.Under this increase in distance, both of these power values would bedecreased by 10⁻⁶.

Development of the low-loss antenna circuit is presented herein to allowfor inductive power coupling at high efficiency over long distances(over 100 meters). To achieve low loss, superconducting materials areused for all current carrying elements, dielectrics are avoided and thesystem is operated at low frequencies (below 200 KHz), and atwavelengths that exceed the antenna diameter by several orders ofmagnitude. Maximum power coupling and maximum efficiency cannot beachieved simultaneously, however efficiencies as high as 50% have beenachieved with the present system at maximum power coupling.

Further reduction in the radiative losses may be achieved by adding anexternal capacitance in which no dielectric is used. To address this,electrical breakdown of a cylindrical capacitor is suppressed by theapplication of a magnetic cross-field that acts to impede the motion ofelectrons across the air gap. The resulting capacitor size is veryreasonable in comparison to the baseline size of the coil. Theinteraction of the system with the environment is quite weak, howevermechanisms for power dissipation, attenuation and modification of thenatural frequency are identified and examined parametrically.

Although this invention has been described in connection with specificforms and embodiments thereof, it will be appreciated that variousmodifications other than those discussed above may be resorted towithout departing from the spirit or scope of the invention as definedin the appended claims. For example, functionally equivalent elementsmay be substituted for those specifically shown and described, certainfeatures may be used independently of other features, and in certaincases, particular locations of elements, steps, or processes may bereversed or interposed, all without departing from the spirit or scopeof the invention as defined in the appended claims.

1. A system for long range wireless energy transfer, comprising: a firstoscillator structure coupled to a power source and at least one secondoscillator structure displaced from said first oscillator structure adistance D to receive energy therefrom, wherein said first and at leastone second oscillator structures are configured into compact coils, saidfirst and said at least one second oscillator structures being formedfrom a superconducting material composition and resonating substantiallyat the same frequency maintained below a predetermined frequency levelsufficient for providing energy coupling between said first and said atleast one second oscillator structures.
 2. The system of claim 1,wherein each of said compact coils of said first and at least one secondoscillator structure is a flat coil having a plurality of windingspositioned in the same plane.
 3. The system of claim 1, wherein saidpredetermined frequency level is below approximately 1 MHz.
 4. Thesystem of claim 1, wherein said predetermined distance D falls in therange covering approximately 10's m and exceeding 100 m.
 5. The systemof claim 1, wherein said power source is coupled up-stream of said firstoscillating structure, said system further including at least one powerconsuming unit coupled down-stream of said at least one secondoscillating unit, a drive coil coupled between said power source andsaid first oscillator structure, and a drain coil coupled between saidat least one second oscillator structure and said at least one powerconsuming unit.
 6. The system of claim 1, wherein diameters of saidcompact coils of said first and said at least one second oscillatorstructures fall in the range between 10 cm and several meters.
 7. Thesystem of claim 1, further comprising a plurality of capacitor elements,each coupled to a respective one of said first and said at least onesecond oscillator structures, said each capacitor element including aninner cylindrical electrode and at least one outer cylindrical electrodedisposed in a co-axial surrounding relationship with said innercylindrical electrode, wherein at least one air gap is defined betweencylindrical walls of said inner and at least one outer cylindricalelectrodes.
 8. The system of claim 7, wherein said each capacitorelement is formed from a superconducting material.
 9. The system ofclaim 8, wherein the superconducting material for said first and atleast one oscillator structures and said each capacitor element isselected from the group consisting of Type I superconductors, HighTemperature Superconductors, including BSCCO, and YBCO, and roomtemperature superconductors.
 10. The system of claim 8, furthercomprising a thermo-control system including a plurality of cryogenicalunits, each operatively engaged with a respective one of said first andat least one second oscillator structures, and said each capacitorelement, to maintain said superconducting material thereof at apredetermined temperature level.
 11. The system of claim 7, wherein eachof said first and at least one second oscillator elements, and said eachcapacitor element is a dielectric-less component.
 12. The system ofclaim 7, wherein said at least one air gap is filled with a dielectricmaterial having a low dissipation factor.
 13. The system of claim 11,further comprising a magnetic field applied axially to said eachcapacitor element to increase a breakdown voltage threshold in said atleast one air gap, thereby increasing the dielectric strength of air insaid at least one air gap of said each dielectric-less capacitorelement.
 14. The system of claim 2, further comprising a dielectricmaterial with a negligible dissipation factor disposed between saidwindings in said flat coils of said first and said at least one secondoscillator structures.
 15. The system of claim 1, further comprising atleast one booster resonator coil positioned between said first and saidat least one second oscillator structures, said at least one boosterresonator coil resonating in phase with said first oscillator structureto receive energy therefrom, and in phase with said at least one secondoscillator structure to transfer energy thereto.
 16. The system of claim7, wherein the length of said each capacitor element l cm≈2 bcm, whereinlcm is the length of said capacitor element, and bcm is the radius ofsaid at least one outer cylindrical electrode in cm.
 17. The system ofclaim 16, wherein the volume of said each capacitor element is${{\pi \cdot b_{cm}^{2} \cdot l_{cm}} \approx {\frac{I_{\max}}{45{\overset{\_}{f} \cdot B_{z}}}\mspace{14mu} {cm}^{3}}},$where Imax is the maximum current of said capacitor element, f is theresonant frequency, and B_(z) is the strength of the axial magneticfield applied to the capacitor element.
 18. A method for long rangewireless energy transfer, comprising the steps of: fabricating first andat least one second oscillator structures as compact flat coils formedfrom a superconducting material, displacing said at least one secondoscillator structure a distance D from said first oscillator structure,coupling said first oscillator structure to a power source, andgenerating an oscillating current of a resonant frequency in said firstoscillator structure, wherein said oscillating current creates anoscillating field, and sensing said oscillating current by said at leastone second oscillator structure, thereby causing oscillation of said atleast one second oscillator structure at said resonant frequency, andthereby transferring energy from said first oscillator structure to saidat least one second oscillator structure, wherein said resonantfrequency is maintained below a predetermined frequency level providinga strong energy coupling between said first and said at least one secondoscillator structures.
 19. The method of claim 18, further comprisingthe steps of: coupling a capacitor element to at least one of said firstand at least one second oscillator structures, said capacitor elementincluding an inner cylindrical electrode and at least one outercylindrical electrode co-axially disposed around said inner cylindricalelectrode, wherein at least one air gap is defined between cylindricalwalls of said inner and at least one outer cylindrical electrodes, andapplying a magnetic field axially to said capacitor element to increasedielectric strength of air in said at least one air gap.
 20. The methodof claim 18, wherein said capacitor element is formed from asuperconducting material, wherein said predetermined frequency level is200 KHz, wherein said distance D exceeds 100 m, and wherein said firstand at least one second oscillator structures and said capacitor elementare dielectric-less components.